This paper is relevant, is full of mathematical insight, and is fun. Consider the unexpected Elton-Odell theorem [J. Elton and E. Odell, Colloq. Math. 44, 105–109 (1981; Zbl 0493.46014)] asserting what its title says or, equivalently, that the so-called Kottman separation constant \(K(X)\) of every infinite-dimensional Banach space is greater than 1. This result provides a true tour-de-force for Riesz’s lemma, and Kottman himself had already shown in [C. A. Kottman, Stud. Math. 53, 15–27 (1975; Zbl 0266.46014)] that the unit ball of every infinite-dimensional Banach space contains some sequence separated by more than one.
In this paper, the author extends the result to symmetrically separated sequences and shows that, for every infinite-dimensional Banach space the symmetric Kottman constant is also greater than one; i.e., \[ K_s(X) = \inf \{\sigma>0: \exists (x_n)\subset B_X,\ \forall n\neq m,\ \|x_n \pm x_m\|\geq \sigma\} >1. \]
The paper is relevant because it closes the problem, and fun because it tells its origin, history and landmarks quite entertainingly (so there is no need to tell the story here: the interested reader is referred to Russo’s paper); it is full of mathematical insight because the two (!) proofs presented are brimful of suggestions and ideas for further research. Perhaps the only ones missing are the “symmetric” versions of the problems mentioned by Diestel in the last two pages of [J. Diestel, Sequences and series in Banach spaces. GTM 92, Springer-Verlag (1984; Zbl 0542.46007)] for the standard Kottman constant.